After a(1) = 2, a(2) = 5, a(4) = 17, a(6) = 41, a(8) = 89, a(10) = 197, and a(12) = 349, which are the next primes? This sequence, which alternates between even and odd, is arbitrary as it is in base 10; what are the analogs in other bases, and the diagonal of the array of such sequences?